To compute the least-squares regression line, we will use the data given for the average prices of a dozen eggs and a gallon of milk. The least-squares regression line is given in the form:
\[ y = mx + b \]
where \( y \) is the dependent variable (price of milk), \( x \) is the independent variable (price of eggs), \( m \) is the slope of the line, and \( b \) is the y-intercept.
Let’s denote:
- \( x_i \) = average price of eggs in month \( i \)
- \( y_i \) = average price of milk in month \( i \)
We are given the following data:
\[ \begin{aligned} \text{Dozen Eggs} & : 1.55, 1.75, 1.63, 1.62, 1.50, 1.37, 1.50, 1.63, 1.83, 1.77 \ \text{Gallon of Milk} & : 3.07, 3.08, 3.07, 3.14, 3.26, 3.43, 3.74, 3.81, 3.84, 3.84 \end{aligned} \]
The formulae needed to calculate \( m \) (slope) and \( b \) (y-intercept) are as follows:
-
Calculate \( n \), the number of data points.
-
Compute sums:
- \( \sum x \)
- \( \sum y \)
- \( \sum xy \)
- \( \sum x^2 \)
- \( \sum y^2 \)
-
Calculate the slope \( m \) using:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]
- Calculate the y-intercept \( b \) using:
\[ b = \frac{\sum y - m(\sum x)}{n} \]
Step-by-step calculations:
-
Count the data points:
- \( n = 10 \)
-
Calculate the sums:
Let's calculate each required sum:
\[ \begin{aligned} \sum x & = 1.55 + 1.75 + 1.63 + 1.62 + 1.50 + 1.37 + 1.50 + 1.63 + 1.83 + 1.77 = 16.10 \ \sum y & = 3.07 + 3.08 + 3.07 + 3.14 + 3.26 + 3.43 + 3.74 + 3.81 + 3.84 + 3.84 = 31.40 \ \sum xy & = (1.55 \cdot 3.07) + (1.75 \cdot 3.08) + (1.63 \cdot 3.07) + (1.62 \cdot 3.14) + (1.50 \cdot 3.26) + (1.37 \cdot 3.43) + (1.50 \cdot 3.74) + (1.63 \cdot 3.81) + (1.83 \cdot 3.84) + (1.77 \cdot 3.84) \ & = 4.7765 + 5.3850 + 4.9991 + 5.0808 + 4.8900 + 4.7031 + 5.6100 + 6.1973 + 7.0368 + 6.7976 \ & = 56.4700\ \sum x^2 & = (1.55^2) + (1.75^2) + (1.63^2) + (1.62^2) + (1.50^2) + (1.37^2) + (1.50^2) + (1.63^2) + (1.83^2) + (1.77^2) \ & = 2.4025 + 3.0625 + 2.6569 + 2.6244 + 2.25 + 1.8769 + 2.25 + 2.6569 + 3.3489 + 3.1329 \ & = 24.0569 \end{aligned} \]
- Calculate the slope \( m \):
\[ \begin{aligned} m & = \frac{10(56.4700) - (16.10)(31.40)}{10(24.0569) - (16.10)^2} \ & = \frac{564.70 - 505.34}{240.569 - 259.21} \ & = \frac{59.36}{-18.641} \ & \approx -3.1852 \end{aligned} \]
- Calculate the y-intercept \( b \):
\[ \begin{aligned} b & = \frac{31.40 - (-3.1852)(16.10)}{10} \ & = \frac{31.40 + 51.34}{10} \ & = \frac{82.74}{10} \ & = 8.2740 \end{aligned} \]
Final least-squares regression line:
The least-squares regression line to predict the price of milk from the price of eggs is:
\[ y = -3.1852x + 8.2740 \]
So, the rounded values are:
- Slope \( m \approx -3.1852 \)
- Y-intercept \( b \approx 8.2740 \)